What is a videogame franchise? What does a franchise mean?
For Castlevania, it means gothic horror action-adventure. Maybe it's a side-scroller, maybe it's in 3D, maybe it's more adventure than straight linear action. Dracula will show up somewhere. But there's some wiggle room.
For Mario, it means silly plumber rescues princess in brightly-coloured worlds with lots of weird gameplay tweaks framed by a largely nonsensical story. Nintendo has done a lot with that formula in the past.
For Final Fantasy, it's very fluid: it means heavy on storytelling, some form of semi-turn-based battles, stat upgrades, and... well, not a whole lot else. A few basic gameplay style standards are pretty much all it means. It also means, though, that it's Square's flagship series and they'll put all they've got into it. Whether it's for better or worse, that's a matter of individual taste, but you know that they've thrown all their guns at it.
For Mega Man, it's almost the total opposite: it's very tight and rigid. Mega Man fights a series of bosses each weak to one other's weapon. Then he uses them all in a series of stages to attack the final boss. Stepping too far away from that strict formula results in lost sales, because the fans want exactly that.
It's a fluid and fluffy description and it really doesn't mean anything more than what anyone takes it to mean, but it's why the opinion on sequels goes so far in all directions. Everyone - marketers, CEOs, programmers, players - seems to have strict and often-conflicting opinions on what a sequel should be. And once a franchise has an identity, it tends to codify a lot of those requirements.
The trick, of course, comes in how a franchise gets that identity. And one game can't create that, no matter how good it is. Sometimes everyone involved (even the players, as mentioned above) are desperate to pigeonhole everything that comes out as soon as possible - perhaps to make it easier to categorise (and either embrace or reject), perhaps to predict how valuable it will be, perhaps other reasons altogether.
The time when a franchise is indelibly associated with a "brand identity", as it were, is hard to pin down, and probably isn't any more specific than "I know it when I see it."